we have an expanding shell of baryons that begins to turn around after decoupling and fall back in to the dark matter well.

At the end of the analysis, one peak is at the origin (somewhat displaced because what's plotted is mass in a spherical shell, not the density), and the other is at the BAO scale.

Why does the BAO peak "stall", instead of ballistically falling back inwards? If I released a spherical shell of non-interacting dark matter particles around a central density, I would expect them to fall back onto the center.

What's the most important physical mechanism preventing that infall? It seems like it can't be the baryon sound speed, because we are (?) looking on scales much larger than the Jeans length. Meanwhile, the spherical symmetry of the problem suggests that any uniform collapse of shell into itself not affect the overall collapse onto the central mass.

It is not an expanding shell of baryons, it is a sound wave. The right picture is not that of an explosion, but that of a plucking of a infinitely long guitar string where the perturbation propagates in both directions (only that now it is in 3d). When universe recombines, photon-baryon plasma ceases to support sound waves, the perturbation then freezes into the correlation function... (the thing is actually a bit more complicated, as the correlation is between origin and 100Mpc where wave has travelled, while in a guitar string, the "perturbation" goes back to zero at the origin, once the delta function perturbation has moved away - I've seen a wrong picture in talks where people show just a ring, without anything at the centre). (But maybe I misunderstood your question)